There are two customs regarding what material to study on Shavu’ot night, both of which are perfectly fine. The first custom is the one established by the kabbalists and called Tikun Leil Shavu’ot. According to this custom, one recites the first three and last three verses of every parsha in the Torah. Sections particularly relevant to the festival, such as those that describe Matan Torah and the Ten Commandments, are recited in their entirety. After the Torah verses, the first and last three verses of each book of the Prophets and the Writings are recited. Afterward, the first and last mishna of every tractate is recited, though some do not recite mishnayot. Afterward all 613 mitzvot are enumerated, followed by midrashim about the giving of the Torah. This is followed by Idra Rabba and other Zohar passages. In addition to those who always follow kabbalistic customs, other communities have adopted this routine as well, and it was the custom of the Vilna Gaon, Ḥatam Sofer, and R. Eliyahu David Rabinowitz-Teomim (Aderet). Some maintain that this order of study should be done with a minyan (Shlah; Ḥida).
The second custom is that each person should learn whatever he wants, as the Sages state: “A person best learns the area of Torah that his heart desires” (AZ 19b). Many yeshiva students study Gemara, as they do during most of their study time. Others choose to study texts related to the value of Torah or the sanctity of the day. It is told that the author of Terumat Ha-deshen, Rabbi Israel Isserlein (who lived about 600 years ago) studied Smak (Sefer Mitzvot Katan) and sometimes Rambam’s Laws of Talmud Torah. Some prefer to study Rambam’s Sefer Ha-mitzvot. Our master Rav Kook would give a lengthy class on Shavu’ot night based on Rambam’s Sefer Ha-mitzvot. Others choose to study topics of interest to them so that they will find it easier to concentrate despite their exhaustion.